On Classical Solutions for Viscous Polytropic Fluids with Degenerate Viscosities and Vacuum

Abstract

In this paper, we consider the three-dimensional isentropic Navier–Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and a vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by a vacuum. Moreover, for certain classes of initial data with a local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.

Appendix: Proof for the Remark 1.4

In this section, we will show that the regular solution that we obtained in Theorem 1.1 is indeed a classical one in $(0, T_*]$. The following lemma will be used in our proof:

Lemma 5.1

[1] If $f(x,t)\in L^2([0,T]; L^2)$, then there exists a sequence $s_k$ such that

$$\begin{aligned} s_k\rightarrow 0, \quad \text {and}\quad s_k |f(x,s_k)|^2_2\rightarrow 0, \quad \text {as} \quad k\rightarrow +\infty . \end{aligned}$$

From the definition of regular solution and the classical Sobolev embedding theorem, it is clear that

$$\begin{aligned} (\rho ,\nabla \rho , \rho _t, u, \nabla u) \in C(\mathbb {R}^3\times [0, T_*]), \end{aligned}$$

so it remains to prove that

$$\begin{aligned} (u_t, \text {div}\mathbb {T})(x,t) \in C(\mathbb {R}^3\times (0, T_*]). \end{aligned}$$

From the proof of Theorem 1.1 in Section 3, we know that (through a change of variable $\phi =\rho ^{\frac{\delta -1}{2}}$), system (1.1) can be written as

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \phi _t+u\cdot \nabla \phi +\frac{\delta -1}{2}\phi \text {div} u=0,\\ \displaystyle u_t+u\cdot \nabla u +\frac{2A\gamma }{\delta -1}\phi ^{\frac{2r-\delta -1}{\delta -1}}\nabla \phi +\phi ^2 Lu=\nabla \phi ^2 \cdot Q(u). \end{array}\right. } \end{aligned}$$

(5.1)

The solution $(\phi ,u)$ satisfies the regularities in (3.6) and $\phi \in C^1(\mathbb {R}^3\times [0, T_*])$.

Step 1 The continuity of $u_t$. We differentiate (5.1)$_2$ with respect to t to get

$$\begin{aligned} \begin{aligned} u_{tt}+\phi ^2 Lu_t=-(\phi ^2)_t Lu-(u\cdot \nabla u)_t -\frac{A\gamma }{\gamma -1}\nabla \Big (\phi ^{\frac{2\gamma -2}{\delta -1}}\Big )_t+(\nabla \phi ^2 \cdot Q(u))_t, \end{aligned} \end{aligned}$$

(5.2)

which, along with (3.6), implies that

$$\begin{aligned} u_{tt}\in L^2([0,T_*];L^2). \end{aligned}$$

(5.3)

Applying the operator $\partial ^\zeta _x$$(|\zeta |=2)$ to (5.2), multiplying the resulting equations by $\partial ^\zeta _x u_t$ and integrating over $\mathbb {R}^3$, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}|\partial ^\zeta _xu_t|^2_2+\alpha |\phi \nabla \partial ^\zeta _x u_t|^2_2+(\alpha +\beta )|\phi \text {div} \partial ^\zeta _x u_t|^2_2\\&\quad =\int \Big ( -\nabla \phi ^2 \cdot \frac{\delta -1}{\delta }Q(\partial ^\zeta _x u_t)-\big (\partial ^\zeta _x(\phi ^2Lu_t)-\phi ^2 L\partial ^\zeta _x u_t\big )\Big )\cdot \partial ^\zeta _x u_t\\&\qquad +\int \Big (-\partial ^\zeta _x\big ((\phi ^2)_t Lu\big )-\partial ^\zeta _x(u\cdot \nabla u)_t -\frac{A\gamma }{\gamma -1}\partial ^\zeta _x\nabla \Big (\phi ^{\frac{2\gamma -2}{\delta -1}}\Big )_t \Big )\cdot \partial ^\zeta _x u_t \\&\qquad +\int \partial ^\zeta _x(\nabla \phi ^2 \cdot Q(u))_t\cdot \partial ^\zeta _x u_t =: \sum _{i=10}^{15}J_i. \end{aligned} \end{aligned}$$

(5.4)

Now we analyze the terms $J_i$$(i=10,\cdots , 15)$. By Hölder’s inequality, Lemma 2.1 and Young’s inequality, we have

$$\begin{aligned} \begin{aligned} J_{10}&=\int \Big ( -\nabla \phi ^2 \cdot \frac{\delta -1}{\delta }Q(\partial ^\zeta _x u_t)\Big )\cdot \partial ^\zeta _x u_t\\&\leqq C|\phi \nabla ^3 u_t|_2|\nabla ^2 u_t|_2|\nabla \phi |_\infty \leqq C| u_t|^2_{D^2}+\frac{\alpha }{20}|\phi \nabla ^3 u_t|^2_2,\\ J_{11}&=\int -\big (\partial ^\zeta _x(\phi ^2Lu_t)-\phi ^2 L\partial ^\zeta _x u_t\big )\cdot \partial ^\zeta _x u_t\\&\leqq C\Big (|\phi \nabla ^3 u_t|_2|\nabla ^2 u_t|_2|\nabla \phi |_\infty +|\nabla \phi |^2_\infty |u_t|^2_{D^2}+|\nabla ^2 \phi |_3|\phi \nabla ^2 u_t|_6|u_t|_{D^2}\Big )\\&\leqq C| u_t|^2_{D^2}+\frac{\alpha }{20}|\phi \nabla ^3 u_t|^2_2,\\ \end{aligned} \end{aligned}$$

(5.5)

and

$$\begin{aligned} J_{12}&=\int -\partial ^\zeta _x\big ((\phi ^2)_t Lu\big )\cdot \partial ^\zeta _x u_t \nonumber \\&\leqq C\Big (|u_t|_{D^2}(|\phi _t|_\infty |\phi \nabla ^4 u|_2+|\nabla ^2 \phi |_3|Lu|_6| \phi _t|_\infty +|\nabla \phi |_\infty |\nabla \phi _t|_6|Lu|_{3})\nonumber \\&\quad +|u_t|_{D^2}(|\phi \nabla ^3 u|_6|\nabla \phi _t|_{3}+|\nabla \phi |_\infty | \phi _t|_\infty |\nabla ^3 u|_{2})+|\phi \nabla ^2 u_t|_6|\phi _t|_{D^2}|Lu|_3\Big ) \nonumber \\&\leqq C\Big (| u_t|^2_{D^2}+|\phi \nabla ^4 u|^2_2+1\Big )+ \frac{\alpha }{20}|\phi \nabla ^3 u_t|^2_2, \nonumber \\ J_{13}&=\int -\partial ^\zeta _x(u\cdot \nabla u)_t \cdot \partial ^\zeta _x u_t \nonumber \\&\leqq C\Vert u_t\Vert _2\Vert u\Vert _3| u_t|_{D^2}+\int -\big (u\cdot \nabla \big ) \partial ^\zeta _x u_t \cdot \partial ^\zeta _x u_t \nonumber \\&\leqq C\Big (1+| u_t|^2_{D^2}+|\nabla u|_\infty |\partial ^\zeta _x u_t|^2_2\Big ) \leqq C\Big (1+| u_t|^2_{D^2}\Big ),\nonumber \\ J_{14}&=\int -\frac{A\gamma }{\gamma -1}\partial ^\zeta _x\nabla \Big (\phi ^{\frac{2\gamma -2}{\delta -1}}\Big )_t \cdot \partial ^\zeta _x u_t=\int \frac{A\gamma }{\gamma -1}\partial ^\zeta _x \Big (\phi ^{\frac{2\gamma -2}{\delta -1}}\Big )_t \text {div} \partial ^\zeta _x u_t \nonumber \\&\leqq C\Big (|\phi ^{\frac{K}{2}-2}|_\infty |\nabla ^2 \phi _t|_2|\phi \nabla ^3 u_t|_2+|\phi ^{\frac{K}{2}-3}|_\infty |\phi _t|_\infty |\nabla ^2 \phi |_2|\phi \nabla ^3 u_t|_2 \nonumber \\&\quad +|\phi ^{\frac{K}{2}-4}|_\infty |\phi _t|_\infty |\nabla \phi |_6|\nabla \phi |_3|\phi \nabla ^3 u_t|_2+|\phi ^{\frac{K}{2}-3}|_\infty |\nabla \phi _t|_2|\nabla \phi |_\infty |\phi \nabla ^3 u_t|_2\Big ) \nonumber \\&\leqq C+\frac{\alpha }{20}|\phi \nabla ^3 u_t|^2_2,\nonumber \\ J_{15}&=\int \partial ^\zeta _x(\nabla \phi ^2 \cdot Q(u))_t\cdot \partial ^\zeta _x u_t \nonumber \\&\leqq C\Big (\Vert \nabla \phi \Vert ^2_2| u_t|^2_{D^2}+\big (\Vert \nabla \phi \Vert _2|\nabla u_t|_3+\Vert u\Vert _3\Vert \phi _t\Vert _2\big )|\phi \nabla ^2 u_t|_6 \nonumber \\&\quad +\big (\Vert \nabla \phi \Vert _2|\phi \nabla ^3 u_t|_2+\Vert \nabla \phi \Vert _2|\phi \nabla ^2 u_t|_6\big )| u_t|_{D^2}\nonumber \\&\quad +\big (\Vert \nabla \phi \Vert _2\Vert \phi _t\Vert _2\Vert u\Vert _3+\Vert \phi _t\Vert _2|\phi \nabla ^3 u|_6\big )| u_t|_{D^2}\Big ) \nonumber \\&\quad +\int \partial ^\zeta _x(\nabla \phi ^2)_t \cdot Q(u)\cdot \partial ^\zeta _x u_t \nonumber \\&\leqq C\Big (| u_t|^2_{D^2}+|\phi \nabla ^4 u|^2_2\Big )+ \frac{\alpha }{20}|\phi \nabla ^3 u_t|^2_2+J_{151}, \end{aligned}$$

(5.6)

where

$$\begin{aligned} \begin{aligned} J_{151}&=\int \partial ^\zeta _x(\nabla \phi ^2)_t \cdot Q(u)\cdot \partial ^\zeta _x u_t\\&\leqq C \Vert \nabla \phi \Vert _2\Vert \phi _t\Vert _2\Vert u\Vert _3|u_t|_{D^2}+\int \phi \partial ^\zeta _x \nabla \phi _t \cdot Q(u)\cdot \partial ^\zeta _x u_t. \end{aligned} \end{aligned}$$

(5.7)

Using integration by parts, the last term in (5.7) is estimated as

$$\begin{aligned} \begin{aligned}&\int \phi \partial ^\zeta _x \nabla \phi _t \cdot Q(u)\cdot \partial ^\zeta _x u_t\\&\quad \leqq C\Big (|\nabla \phi |_\infty |\phi _t|_{D^2}|\nabla u|_\infty |u_t|_{D^2}+|\phi _t|_{D^2}|\nabla ^2u|_{3}|\phi \nabla ^2 u_t|_6\\&\qquad +|\phi \nabla ^3 u_t|_2|\nabla u|_\infty \Vert \phi _t\Vert _2\Big )\\&\quad \leqq C| u_t|^2_{D^2}+\frac{\alpha }{20}|\phi \nabla ^3 u_t|^2_2. \end{aligned} \end{aligned}$$

(5.8)

Then (5.4) reduces to

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}|u_t|^2_{D^2}+\frac{\alpha }{2} |\phi \nabla ^3 u_t|^2_2 \leqq C\Big (| u_t|^2_{D^2}+|\phi \nabla ^4 u|^2_2+1\Big ). \end{aligned} \end{aligned}$$

(5.9)

Multiplying both sides of (5.9) with s and integrating the resulting inequalities over $[\tau ,t]$ for any $\tau \in (0,t)$, we have

$$\begin{aligned} \begin{aligned}&t|u_t|^2_{D^2}+\int _{\tau }^t s|\phi \nabla ^3 u_t|^2_2 \,\mathrm{d}s \leqq C\tau | u_t(\tau )|^2_{D^2}+C(1+t). \end{aligned} \end{aligned}$$

(5.10)

According to the definition of the regular solution, we know that

$$\begin{aligned} \nabla ^2 u_t \in L^2([0,T_*]; L^2). \end{aligned}$$

Using Lemma 5.1 to $\nabla ^2 u_t$, there exists a sequence $s_k$ such that

$$\begin{aligned} s_k\rightarrow 0, \quad \text {and}\quad s_k |\nabla ^2 u_t(\cdot , s_k)|^2_2\rightarrow 0, \quad \text {as} \quad k\rightarrow +\infty . \end{aligned}$$

Choosing $\tau =s_k \rightarrow 0$ in (5.10), we have

$$\begin{aligned} \begin{aligned}&t|u_t|^2_{D^2}+\int _{0}^t s|\phi \nabla ^3 u_t|^2_2 \,\mathrm{d}s \leqq C(1+t), \end{aligned} \end{aligned}$$

(5.11)

then

$$\begin{aligned} t^{\frac{1}{2}}u_t \in L^\infty ([0,T_*]; H^2). \end{aligned}$$

(5.12)

The classical Sobolev embedding theorem gives

$$\begin{aligned} \begin{aligned} L^\infty ([0,T];H^1)\cap W^{1,2}([0,T];H^{-1})\hookrightarrow C([0,T];L^q) \end{aligned} \end{aligned}$$

(5.13)

for any $q\in (3,6]$. From (5.3) and (5.12) we have

$$\begin{aligned} tu_t \in C([0,T_*];W^{1,4}), \end{aligned}$$

which implies that

$$\begin{aligned} u_t \in C(\mathbb {R}^3\times (0,T_*] ). \end{aligned}$$

Step 2 The continuity of $\text {div}\mathbb {T}$. Denote $\mathbb {N}=\phi ^2 Lu-\nabla \phi ^2 \cdot Q(u)$. From equations (5.1)$_2$, regularities (3.6) and (5.12), it is easy to show that

$$\begin{aligned} t\mathbb {N} \in L^\infty ([0,T_*]; H^2). \end{aligned}$$

Due to

$$\begin{aligned} \mathbb {N}_t \in L^2([0,T_*]; L^2), \end{aligned}$$

we obtain from (5.13) that

$$\begin{aligned} t \mathbb {N}\in C([0,T_*];W^{1,4}), \end{aligned}$$

which implies that

$$\begin{aligned} \mathbb {N} \in C(\mathbb {R}^3\times (0, T_*]). \end{aligned}$$

Since $\rho \in C(\mathbb {R}^3\times [0, T_*])$ and $\text {div}\mathbb {T}=\rho \mathbb {N}$, we immediately obtain the desired conclusion.

In summary, we have shown that the regular solution that we obtained is indeed a classical one in $\mathbb {R}^3\times [0, T_*]$ to the Cauchy problem (1.1)–(1.3).